The Algebra of Logic Tradition

The algebra of logic, as an explicit algebraic
system showing the underlying mathematical structure of logic, was
introduced by George Boole (1815–1864) in his book The
Mathematical Analysis of Logic (1847). The methodology initiated
by Boole was successfully continued in the 19th century in
the work of William Stanley Jevons (1835–1882), Charles Sanders
Peirce (1839–1914), Ernst Schröder (1841–1902), among many
others, thereby establishing a tradition in (mathematical) logic. From
Boole's first book until the influence after WWI of the monumental
work Principia Mathematica (1910–1913) by Alfred North
Whitehead (1861–1947) and Bertrand Russell (1872–1970),
versions of the algebra of logic were the most developed form of
mathematical logic above all as presented in Schröder's three volumes
Vorlesungen über die Algebra der Logik (1890–1905).
Furthermore, this tradition motivated the investigations of Leopold
Löwenheim (1878–1957) that eventually gave rise to model theory.
In addition, in 1941, Alfred Tarski (1901–1983) in his paper
“On the calculus of relations” returned to Peirce's
relation algebra as presented in Schröder's Algebra der
Logik. The tradition of the algebra of logic played a key role in
the notion of Logic as Calculus as opposed to the notion of
Logic as Universal Language. Beyond Tarski's algebra of
relations, the influence of the algebraic tradition in logic can be
found in other mathematical theories, such as category theory. However
this influence lies outside the scope of this entry, which is divided
into 10 sections.

Boole's The Mathematical Analysis of Logic presents many
interesting logic novelties: It was the beginning of
nineteenth-century mathematization of logic and provided an
algorithmic alternative (via a slight modification of
ordinary algebra) to the catalog approach used in traditional
logic (even if reduction procedures were developed in the
latter). That is, it provided an effective method for proving logical
laws on the basis of a system of postulates. As Boole wrote later, it
was a proper “science of reasoning”, and not a
“mnemonic art” like traditional Syllogistics (Boole 1997:
136). Three-quarters of the way through this book, after finishing
his discussion of syllogistic logic, Boole started to develop the
general tools that would be used in his Laws of Thought
(1854) to greatly extend traditional logic by permitting an argument
to have many premises and to involve many classes. To handle the
infinitely many possible logical arguments of this expanded logic, he
presented theorems that provided key tools for an algorithmic analysis
(a catalog was no longer feasible).

Boole's ideas were conceived completely independently of earlier
anticipations, like those developed by G.W. Leibniz. However, they
emerged from the particular contexts of English mathematics (see
Peckhaus 2009). According to Víctor Sánchez Valencia, the tradition
that originated with Boole came to be known as the algebra of
logic since the publication in 1879 of Principles of the
Algebra of Logic by Alexander MacFarlane (see Sánchez Valencia
2004: 389). MacFarlane considered “the analytical method of
reasoning about Quality proposed by Boole” as an algebra (see
MacFarlane 1879: 3).

This approach differs from what is usually called algebraic
logic, understood as

a style [of
logic] in which concepts and relations are expressed by
mathematical symbols [\(\ldots\)] so that mathematical techniques can be
applied. Here mathematics shall mean mostly algebra, i.e., the part of
mathematics concerned with finitary operations on some set. (Hailperin
2004: 323)

Algebraic logic can be already found in the work of Leibniz, Jacob
Bernoulli and other modern thinkers, and it undoubtedly constitutes
an important antecedent of Boole's approach. In a broader perspective,
both are part of the tradition of
symbolic knowledge in the formal sciences, as first conceived
by Leibniz (see Esquisabel 2012). This idea of algebraic logic was
continued to some extent in the French Enlightenment in the work of
Condillac and Condorcet (see Grattan-Guinness 2000: 14 ff.)

Boole's methodology for dealing with logical problems can be
described as follows:

Translate the logical data into suitable equations;

Apply algebraic techniques to solve these equations;

Translate this solution, if possible, back into the original language.

Later, in his Pure Logic from 1864, Jevons changed Boole's
partial operation of union of disjoint sets to the modern unrestricted
union and eliminated Boole's questionable use of uninterpretable terms
(see Jevons 1890). Peirce (1880) eliminated explicitly the
Aristotelian derivation of particular statements from universal
statements by giving the modern meaning for “All \(A\) is
\(B\)”. In addition, he extended the algebra of logic for classes
to the algebra of logic for binary relations and introduced general
sums and products to handle quantification. Ernst Schröder,
taking inspiration from previous work by Hermann (1809–1877) and
Robert Grassmann (1815–1901) and using the framework developed
by Peirce, developed and systematized the 19th Century
achievements in the algebra of logic in his three-volume work
Vorlesungen über die Algebra der Logik (1890–1910).

The contributions of Gottlob Frege (1848–1925) to logic from
the period 1879–1903, based on an axiomatic approach to
logic, had very little influence at the time (and the same can be said
of the diagrammatic systems of C.S. Peirce developed at the
turn of the century). Whitehead and Russell rejected the algebra of
logic approach, with its predominantly equational formulations and
algebraic symbolism, in favor of an approach strongly inspired by the
axiomatic system of Frege, and using the notation developed by
Giuseppe Peano, namely to use logical connectives, relation symbols
and quantifiers.

During the first two decades of the twentieth century, the algebra
of logic was further developed in the works of Poretzsky, Luis Couturat,
Leopold Löwenheim, and Heinrich Behmann. In particular, elimination
theorems in the algebra of logic influenced decision procedures for
fragments of first-order and second-order logic (see Mancosu, Zach,
Badesa 2009).

After WWI David Hilbert (1862–1943), who had at first adopted
the algebraic approach, picked up on the approach of
Principia, and the algebra of logic fell out of
favor. However, in 1941, Tarski treated relation algebras as an
equationally defined class. Such a class has many models besides the
collection of all binary relations on a given universe that
was considered in the 1800s, just as there are many Boolean algebras
besides the power set Boolean algebras studied in the
1800s. In the years 1948–1952 Tarski, along with his students
Chin and Thompson, created cylindric algebras as an algebraic logic
companion to first-order logic, and in 1956 Paul Halmos
(1916–2006) introduced polyadic algebras for the same
purpose. As Halmos (1956 b, c and d)
noted, these new algebraic logics tended to focus on studying the
extent to which they captured first-order logic and on their universal
algebraic aspects such as axiomatizations and structure theorems, but
offered little insight into the nature of the first-order logic which
inspired their creation.

In late 1847, Boole and Augustus De Morgan (1806–1871) each
published a book on logic—Boole's Mathematical Analysis of
Logic (1847) and De Morgan's Formal Logic (1847). De
Morgan's approach was to dissect every aspect of traditional deductive
logic (usually called ‘Aristotelian logic’) into its
minutest components, to consider ways to generalize these components,
and then, in some cases, undertake to build a logical system using
these components. Unfortunately, he was never able to incorporate his
best ideas into a significant system. His omission of a symbol for
equality made it impossible to develop an equational algebra of
logic. It seems that synthesis was not De Morgan's strong suite.

Boole approached logic from a completely different perspective,
namely how to cast Aristotelian logic in the garb of symbolical
algebra. Using symbolical algebra was a theme with which he was
well-acquainted from his work in differential equations, and from the
various papers of his young friend and mentor Duncan Farquharson Gregory (1813–1844), who made
attempts to cast other subjects such as geometry into the language of
symbolical algebra. Since the application of symbolical algebra to
differential equations had proceeded through the introduction of
differential operators, it must have been natural for Boole to look
for operators that applied in the area of Aristotelian logic. He
readily came up with the idea of using “selection”
operators, for example, a selection operator for the color red would
select the red members from a class. In his 1854 book, Boole realized
that it was simpler to omit selection operators and work directly with
classes. (However he kept the selection operators to justify his
claim that his laws of logic were not ultimately based on observations
concerning the use of language, but were actually deeply rooted in the
processes of the human mind.) From now on in this article, when
discussing Boole's 1847 book, the selection operators have been
replaced with the simpler direct formulation using classes.

Since symbolical algebra was just the syntactic side of ordinary
algebra, Boole needed ways to interpret the usual operations and
constants of algebra to create his algebra of logic for classes.
Multiplication was interpreted as intersection, leading to his one new
law, the idempotent law \(XX = X\) for multiplication, rediscovering a
logical law already formulated by Leibniz. Addition was defined as
union, provided one was dealing with disjoint classes; and
subtraction as class difference, provided one was subtracting a
subclass from a class. In other cases, the addition and subtraction
operations were simply undefined, or as Boole wrote,
uninterpretable. The usual laws of arithmetic told Boole that
1 must be the universe and \(1 - X\) must be the complement of \(X\).

The next step in Boole's system was to translate the four kinds of
categorical propositions into equations, for example “All \(X\) is
\(Y\)” becomes \(X = XY\), and “Some \(X\) is \(Y\)” becomes
\(V = XY\), where \(V\) is a new symbol. To eliminate the middle term in a
syllogism Boole borrowed an elimination theorem from ordinary algebra,
but it was too weak for his algebra of logic. This would be remedied
in his 1854 book. Boole found that he could not always derive the
desired conclusions with the above translation of particular
propositions (i.e., those with existential import), so he added the
variants \(X = VY\), \(Y = VX\), and \(VX = VY\) (see the entry on
Boole).

The symbolic algebra of the 1800s included much more than just the
algebra of polynomials, and Boole experimented to see which results
and tools might apply to the algebra of logic. For example, he proved
one of his results by using an infinite series expansion. His
fascination with the possibilities of ordinary algebra led him to
consider questions such as: What would logic be like if the idempotent
law were replaced by the law \(X^3 = X\)? His successors, especially
Jevons, would soon narrow the operations on classes to the ones that
we use today, namely union, intersection and complement.

As mentioned earlier, three-quarters of the way through his brief
book of 1847, after finishing derivations of the traditional
Aristotelian syllogisms in his system, Boole announced that his
algebra of logic was capable of far more general applications. Then he
proceeded to add general theorems on developing (expanding) terms,
providing interpretations of equations, and using long division to
express one class in an equation in terms of the other classes (with
side conditions added).

Boole's theorems, completed and perfected in 1854, gave algorithms
for analyzing infinitely many argument forms. This opened a new and
fruitful perspective, deviating from the traditional approach to
logic, where for centuries scholars had struggled to come up with
clever mnemonics to memorize a very small catalog of valid conversions
and syllogisms and their various interrelations.

De Morgan's Formal Logic did not gain significant
recognition, primarily because it was a large collection of small
facts without a significant synthesis. Boole's The Mathematical
Analysis of Logic had powerful methods that caught the attention
of a few scholars such as De Morgan and Arthur Cayley
(1821–1895); but immediately there were serious questions about
the workings of Boole's algebra of logic: Just how closely was it tied
to ordinary algebra? How could Boole justify the procedures of his
algebra of logic? In retrospect it seems quite certain that Boole did
not know why his system worked. His claim, following Gregory, that in
order to justify using ordinary algebra it was enough to check the
commutative law \(XY = YX\) for multiplication and the distributive law
\(X(Y + Z) = XY + XZ\), is clearly false. Nonetheless it is also likely
that he had checked his results in a sufficient number of cases to
give substance to his belief that his system was correct.

In his second book, The Laws of Thought, Boole not only
applied algebraic methods to traditional logic but also attempted some
reforms to logic. He started by augmenting the laws of his 1847
algebra of logic (without explicitly saying that his previous list of
three axioms was inadequate), and made some comments on the rule of
inference (performing the same operation on both sides of an
equation). But then he casually stated that the foundation of his
system actually rested on a single (new) principle, namely it sufficed
to check an argument by seeing if it was correct when the class
symbols took on only the values 0 and 1, and the operations were the
usual arithmetical operations. Let us call this Boole's Rule of 0
and 1. No meaningful justification was given for Boole's adoption
of this new foundation, it was not given a special name, and the scant
references to it in the rest of the book were usually rather clumsily
stated. For a modern analysis of this Rule of 0 and 1 see Burris &
Sankappanavar 2013.

The development of the algebra of logic in the Laws of
Thought proceeded much as in his 1847 book, with minor changes to
his translation scheme, and with the selection operators replaced by
classes. There is a new and very important theorem (correcting the one
he had used in 1847), the Elimination Theorem, which says the
following: given an equation \(F(x,y, z, \ldots) = 0\) in the class
symbols \(x, y, z\), etc., the most general conclusion that follows from
eliminating certain of the class symbols is obtained by (1)
substituting 0s and 1s into \(F(x, y, z, \ldots)\) for the symbols to be
eliminated, in all possible ways, then (2) multiplying these various
substitution instances together and setting the product equal to
0. Thus eliminating \(y\) and \(z\) from \(F(x, y, z) = 0\) gives \(F(x, 0,
0)F(x, 0, 1)F(x, 1, 0)F(x, 1, 1) = 0\). This theorem also played an
important role in Boole's interpretation of Aristotle's
syllogistic.

From an algebra of logic point of view, the 1854 treatment at times
seems less elegant than that in the 1847 book, but it gives a much
richer insight into how Boole thinks about the foundations for his
algebra of logic. The final chapter on logic, Chapter XV, was an
attempt to give a uniform proof of the Aristotelian conversions and
syllogisms. (It is curious that prior to Chapter XV Boole did not
present any examples of arguments involving particular propositions.)
The details of Chapter XV are quite involved, mainly because of the
increase in size of expressions when the Elimination and Development
Theorems are applied. Boole simply left most of the work to the
reader. Later commentators would gloss over this chapter, and no one
seems to have worked through its details.

Aside from the Rule of 0 and 1 and the Elimination Theorem, the
1854 presentation is mainly interesting for Boole's attempts to
justify his algebra of logic. He argued that in symbolical algebra it
was quite acceptable to carry out equational deductions with partial
operations, just as one would when the operations were total, as long
as the terms in the premises and the conclusion were interpretable. He
said this was the way ordinary algebra worked with the uninterpretable
\(\sqrt{-1}\), the square root of −1. (The geometric
interpretation of complex numbers was recognized early on by Wessel,
Argand, and Gauss, but it was only with the publications of Gauss and
Hamilton in the 1830s that doubts about the acceptability of complex
numbers in the larger mathematical community were overcome. It is
curious that in 1854 Boole regarded \(\sqrt{-1}\) as
uninterpretable.)

There were a number of concerns regarding Boole's approach to the
algebra of logic:

Was there a meaningful tie between his algebra of logic and the
algebra of numbers, or was it just an accident that they were so
similar?

Could one handle particular propositions in an algebraic logic that
focused on equations?

Was it really acceptable to work with uninterpretable terms in
equational derivations?

Was Boole using “Aristotelian” semantics (the semantics
presupposed in traditional logic, where the extension of a term is
nonempty)?

Jevons, who had studied with De Morgan, was the first to offer an
alternative to Boole's system. In 1863 he wrote to Boole that surely
Boole's operation of addition should be replaced by the more natural
‘inclusive or’ (or ‘union’), leading to the
law \(X + X = X\). Boole completely rejected this suggestion (it would
have destroyed his system based on ordinary algebra) and broke off the
correspondence. Jevons published his system in his 1864 book,
Pure Logic (reprinted in Jevons 1890). By ‘pure’ he meant
that he was casting off any dependence on the algebra of
numbers—instead of classes, which are associated with quantity,
he would use predicates, which are associated with quality, and his
laws would be derived directly from the (total) fundamental operations
of inclusive disjunction and conjunction. But he kept Boole's use of
equations as the fundamental form of statements in his algebra of
logic.

By adopting De Morgan's convention of using upper-case/lower-case
letters for complements, Jevons' system was not suited to provide
equational axioms for modern Boolean algebra. However, he refined his
system of axioms and rules of inference until the result was
essentially the modern system of Boolean algebra for ground
terms, that is, terms where the class symbols are to be thought
of as constants, not as variables.

It must be noticed that modern equational logic deals with
universally quantified equations (which would have been
called laws in the 1800s). In the 19th century
algebra of logic one could translate “All \(X\) is \(Y\)” as
the equation \(X = XY\). This is not to be viewed as the
universally quantified expression \((''X)(''Y)(X = XY)\). \(X\) and \(Y\)
are to be treated as constants (or schematic letters). Terms that only
have constants (no variables) are called ground terms.

By carrying out this analysis in the special setting of an algebra
of predicates (or equivalently, in an algebra of classes) Jevons
played an important role in the development of modern equational
logic. As mentioned earlier, Boole gave inadequate sets of equational
axioms for his system, originally starting with the two laws due to
Gregory plus his idempotent law;
these were accompanied by De Morgan's inference rule that one could
carry out the same operation (Boole's fundamental operations in his
algebra of logic were addition, subtraction and multiplication) on
equals and obtain equals. Boole then switched to the simple and
powerful (but unexplained) Rule of 0 and 1.

Having replaced Boole's fundamental operations with total
operations, Jevons proceeded, over a period of many years, to work on
the axioms and rules for his system. Some elements of equational logic
that we now take for granted required a considerable number of years
for Jevons to resolve:

The Reflexive Law (\(A=A\)). In 1864 Jevons listed this as a
postulate (1890, p. 11) and then in §24 he referred to \(A = A\) as a
“useless Identical proposition”. In his 1869 paper on
substitution it became the “Law of Identity”. In the
Principles of Science (1874) it was one of the three
“Fundamental Laws of Thought”.

The Symmetric Law (\(B = A\) follows from \(A = B\)). In 1864
Jevons wrote “\(A = B\) and \(B = A\) are
the same statement”. This is a position he would maintain. In
1874 he wrote

I shall consider the two forms \(A =
B\) and \(B = A\) to express exactly the same identity written
differently.

For a final form of his algebra of logic we turn to the laws which
he had scattered over 40 pages in Principles of Science
(1874), having replaced his earlier use of + by
\(\ORjev\), evidently to move
further away from any appearance of a connection with the algebra of
numbers:

For his single rule of inference Jevons chose his principle of
substitution—in modern terms this was essentially a combination
of ground replacement and transitivity. He showed how to derive
transitivity of equality from this; he could have derived symmetry as
well but did not. The associative law was missing—it was
implicit in the lack of parentheses in his expressions.

It was only in his Studies in Deductive Logic (1880) that
Jevons mentioned McColl's use of an accent to indicate negation. After
noting that McColl's accent allowed one to take the negation of
complex bracketed terms he went on to say that, for the most part, he
found the notation of De Morgan, the notation that he had always used,
to be the more elegant.

Peirce started his research into the algebra of logic in the late
1860s. In his paper “On an Improvement in Boole's calculus of
Logic” (Peirce 1867), he arrived
independently at the same conclusion that Jevons had reached earlier,
that one needed to replace Boole's partial operation of addition with
the total operation of union (see CP 3.3.6). In his important 1880
paper, “On the Algebra of Logic”, Peirce quietly broke
with the traditional extensional semantics and introduced a usual
assumption of modern semantics: the extension of a concept, understood
as a class, could be empty (as well as the universe), and stated the
truth values of the categorical propositions that we use today. For
example, he said the proposition “All \(A\) is \(B\)” is true
if \(A\) and \(B\) are both the empty class. Conversion by Limitation,
that is, the argument “All \(A\) is \(B\)” therefore
“Some \(B\) is \(A\)”, was no longer a valid inference. Peirce
said nothing about the reasons for and merits of his departure from
the traditional semantic assumption of existence.

Peirce also broke with Boole's and Jevons' use of equality as the
fundamental primitive, using instead the relation of
“subsumption” interpretable in different ways (subclass
relation, implication, etc.). He stated the partial order properties
of subsumption and then proceeded to define the operations of + and
× as least upper bounds and greatest lower bounds—he
implicitly assumed such bounds existed—and listed the key
equational properties of the algebras with two binary operations that
we now call lattices. Then he claimed that the distributive law
followed, but said the proof was too tedious to include. The
fruitfulness of this perspective is evident in his seminal paper from
1885. There Peirce introduced a system for propositional logic based
on five axioms for implication (represented by the sign
‘\(-\kern-.4em<\)’), including what is now called
Peirce's law. It certainly made the algebra of logic more elegant.

De Morgan wrote a series of six papers called “On the
Syllogism” in the years 1846 to 1863 (reprinted in De Morgan
1966). In his efforts to generalize the syllogism, De Morgan replaced
the copula “is” with a general binary relation in the
second paper of the series dating from 1850. By allowing different
binary relations in the two premises of a syllogism, he was led to
introduce the composition of the two binary relations to express the
conclusion of the syllogism. In this pursuit of generalized syllogisms
he introduced various other operations on binary relations, including
the converse operation, and he developed a fragment of a calculus for
these operations. His main paper on this subject was the fourth in the
series, called “On the syllogism, No. IV, and on the logic of
relations” published in 1859 (see De Morgan 1966).

Following De Morgan's paper, Peirce, in his paper
“Description of a Notation for the Logic of Relatives, resulting
from an Amplification of the Conceptions of Boole's Calculus of
Logic” from 1870, lifted Boole's work to the setting of binary
relations—with binary relations one had, in addition to union,
intersection and complement, the natural operations of composition and
converse. A binary relation was characterized as a set of ordered
pairs (see 3.328). He worked on this new calculus between 1870 and
1883. Like De Morgan, Peirce also considered a number of other natural
operations on relations. Peirce's main paper on the subject was
“On the Algebra of Logic” (1880). By employing
unrestricted unions, denoted by Σ, and unrestricted
intersections, denoted by Π, Peirce thus introduced quantifiers
into his algebra of logic.

In a paper from 1882, “Brief Description of the Algebra of
Relatives”, reprinted in De Morgan 1966, he used these
quantifiers to define operations on relations by means of operations
on certain kind of coefficients. De Morgan gets credit for introducing
the concept of relation, but Peirce is considered the true creator of
the theory of relations (see, e.g., Tarski 1941: 73). However, Peirce
did not develop this theory. As Calixto Badesa wrote, “the
calculus of relatives was never to Peirce's liking” (Badesa
2004: 32). He considered it too complicated because of the combination
of class operations with relational ones. Instead, he preferred from
1885 onwards to develop a “general algebra” including
quantifiers but no operation on relations. In this way, he arrived at
an elementary and informal presentation of what is now called
first-order logic (see Badesa 2004,
loc. cit.).

The German mathematician Ernst Schröder played a key role in the
tradition of the algebra of logic. A good example was his challenge to
Peirce to provide a proof of the distributive law, as one of the key
equational properties of the algebras with two binary operations.
Peirce (1885) admitted that he could not provide a proof. Years later
Huntington (1904: 300–301) described part of the content of a
letter he had received from Peirce in December 1903 that claimed to
provide the missing proof—evidently Peirce had stumbled across
the long lost pages after the death of Schröder in 1902. Peirce
explained to Huntington that he had originally assumed Schröder's
challenge was well-founded and that this apparent shortcoming of his
paper “was to be added to the list of blunders, due to the
grippe, with which that paper abounds, …”. Actually
Peirce's proof did not correct the error since the distributive law
does not hold in lattices in general; instead his proof brought in the
operation of complementation—he used the axiom

if \(a\) is not contained in the complement of \(b\)
then \(a\) and \(b\) have a common lower bound.

On the basis of his previous algebraic work, Schröder wrote an
encyclopedic three volume work at the end of the 19th
century called Vorlesungen über die Algebra der Logik
(1890–1905), built on the subsumption framework with the modern
semantics of classes as presented by Peirce. This work was the result
of his research in algebra and revealed different influences.
Schröder aimed at a general algebraic theory with applications in many
mathematical fields, where the algebra of logic was at the core. As
Geraldine Brady pointed out, it offers the first exposition of
abstract lattice theory, the first exposition of Dedekind's theory of
chains after Dedekind, the most comprehensive development of the
calculus of relations, and a treatment of the foundations of
mathematics on the basis of the relation calculus (see Brady 2000: 143
f.)

The first volume concerned the equational logic of classes, the
main result being Boole's Elimination Theorem of 1854. Three rather
complicated counter-examples to Peirce's claim of distributivity
appeared in an appendix to Vol. I, one of which involved nine-hundred
and ninety identities for quasigroups. On the basis of this volume,
Dedekind (1897) composed an elegant modern abstract presentation of
lattices (which he called Dualgruppen); in this paper he
presented a five-element counter-example to Peirce's claim of the
distributive law.

Volume II augments the algebra of logic for classes developed in
Volume I so that it can handle existential statements. First, using
modern semantics, Schröder proved that one cannot use equations to
express “Some \(X\) is \(Y\)”. However, he noted that one can
easily express it with a negated equation, namely \(XY \ne 0\). Volume
II, a study of the calculus of classes using both equations and
negated equations, attempted to cover the same topics covered in
Vol. I, in particular there was considerable effort devoted to finding
an Elimination Theorem. After dealing with several special cases,
Schröder recommended this topic as an important research
area—the quest for an Elimination Theorem would be known as the
Elimination Problem.

Inspired mainly by Peirce's work, Schröder examined the algebra of
logic for binary relations in Vol. III of his Vorlesungen über die
Algebra der Logik. As Tarski once noted, Peirce's work was
continued and extended in a very thorough and systematic way by
Schröder. One item of particular fascination for him was this: given
an equation \(E(x, y, z, \ldots) = 0\) in this algebra, find the general
solution for one of the relation symbols, say for \(x\), in terms of the
other relation symbols. He managed, given a particular solution \(x =
x0\), to find a remarkable term \(S(t, y, z, \ldots)\) with the following
properties: (1) \(x = S(t, y, z, \ldots)\) yields a solution to \(E = 0\)
for any choice of relation \(t\), and (2) every solution \(x\) of \(E = 0\)
can be obtained in this manner by choosing a suitable \(t\). Peirce was
not impressed by Schröder's preoccupation with the problem of solving
equations, and pointed out that Schröder's parametric solution was a
bit of a hoax—the expressive power of the algebra of logic for
relations was so strong that by evaluating the term \(S(t, y, z,
\ldots)\) one essentially carried out the steps to check if \(E(t, y, z,
\ldots) = 0\); if the answer was yes then \(S(t, y, z, \ldots)\) returned
the value \(t\), otherwise it would return the value \(x0\).

Summing up, Schröder constructed an algebraic version of modern
predicate logic and also a theory of relations. He applied it to
different fields (e.g., Cantor's set theory), and he considered his
algebraic notation as a general or universal language
(pasigraphy, see Peckhaus 2004 and Legris 2012). It is to be
noted that Löwenheim in 1940 still thought it was as reasonable as set
theory. According to him, Schröder's idea of solving a relational
equation was a precursor of Skolem functions, and Schröder inspired
Löwenheim's formulation and proof of the famous theorem that every
“arithmetical” sentence with an infinite model has a
countable model. Schröder's calculus of relations was the basis for
the doctoral dissertation of Norbert Wiener (1894–1964) in
Harvard (Wiener 1913). According to Brady, Wiener gave the first
axiomatic treatment of the calculus of relations, preceding Tarski's
axiomatization by more than twenty years (see Brady 2000: 165).

At the turn of the 19th Century, David Hilbert
(1862–1943) presented, in his Grundlagen der Geometrie,
Euclidean geometry as an axiomatic subject that did not depend on
diagrams for its proofs (Hilbert 1899). This led to a wave of interest
in studying axiom systems in mathematics; in particular one wanted to
know if the axioms were independent, and which primitives led to the
most elegant systems. Edward Vermilye Huntington (1874–1952) was
one of the first to examine this issue for the algebra of logic. He
gave three axiomatizations of the algebra of logic, showed each set of
axioms was independent, and that they were equivalent (see Huntington
1904). In 1933 he returned to this topic with three new sets of
axioms, one of which contained the following three equations (1933:
280):

Shortly after this, Herbert Robbins (1915–2001) conjectured
that the third equation could be replaced by the slightly simpler

\[
[(a + b)' + (a + b')']' = a.
\]

Neither Huntington nor Robbins could prove this, and later it
withstood the efforts of many others, including even Tarski and his
talented school at Berkeley. Building on partial results of Winker,
the automated theorem prover EQP, designed by William McCune of the
Argonne National Laboratory, found a proof of the Robbins Conjecture
in 1996. This accomplishment was popularized in Kolata 2010.

According to Huntington (1933: 278), the term “Boolean
algebra” was introduced by Henry M. Sheffer (1882–1964) in
the paper where he showed that one could give a five-equation
axiomatization of Boolean algebra using the single fundamental
operation of joint exclusion, now known as the Sheffer stroke (see
Sheffer 1913). Whitehead and Russell claimed in the preface to the
second edition of Principia that the Sheffer stroke was the greatest
advance in logic since the publication of Principia. (Hilbert
and Ackermann (1928), by contrast, stated that the Sheffer stroke was
just a curiosity.) Neither realized that decades earlier Schröder had
discovered that the dual of the Sheffer stroke was also such an
operation—Schröder's symbol for his operation was that of a
double-edged sword.

In the 1930s Garrett Birkhoff (1911–1996) established the
fundamental results of equational logic, namely (1) equational classes
of algebras are precisely the classes closed under homomorphisms,
subalgebras and direct products, and (2) equational logic is based on
five rules: reflexivity, symmetry, transitivity, replacement, and
substitution. In the 1940s, Tarski joined in this development of
equational logic; the subject progressed rapidly from the 1950s till
the present time.

Traditional logic studied certain simple relationships between
classes, namely being a subclass of and having a nonempty
intersection with. However, once one adopted an axiomatic
approach, the topic of possible models besides the obvious ones
surfaced. Beltrami introduced models of non-Euclidean geometry in the
late 1860s. In the 1890s Schröder and Dedekind constructed models of
the axioms of lattice theory to show that the distributive law did not
follow. But when it came to the algebra of classes, Schröder
considered only the standard models, namely each was the collection of
all subclasses of a given class.

The study of general models of the axioms of Boolean algebra did
not get underway until the late 1920s; it was soon brought to a very
high level in the work of Marshall Harvey Stone (1903–1989) (see
his papers 1936, 1937). He was interested in the structure of rings of
linear operators and realized that the central idempotents, that is,
the operators \(E\) that commuted with all other operators in the ring
under multiplication (that is, \(EL = LE\) for all \(L\) in the ring) and
which were idempotent under multiplication (\(EE = E\)) played an
important role. In a natural way, the central idempotents formed a
Boolean algebra.

Pursuing this direction of research led Stone to ask about the
structure of an arbitrary Boolean algebra, a question that he answered
by proving that every Boolean algebra is isomorphic to a Boolean
algebra of sets. In his work on Boolean algebras he noticed a
certain analogy between kernels of homomorphisms and the ideals
studied in ring theory—this led him to give the name
“ideal” to such kernels. Not long after this he discovered
a translation between Boolean algebras and Boolean rings; under this
translation the ideals of a Boolean algebra corresponded precisely to
the ideals of the associated Boolean ring. His next major contribution
was to establish a correspondence between Boolean algebras and certain
topological spaces now called Boolean spaces (or Stone spaces). This
correspondence would later prove to be a valuable tool in the
construction of exotic Boolean algebras. These results of Stone are
still a paradigm for developments in the algebra of logic.

Inspired by the rather brief treatment of first-order statements
about relations in Vol. III of the Algebra der Logik,
Löwenheim (1915) showed that if such a statement could be satisfied in
an infinite domain then it could be satisfied in a denumerable
domain. In 1920 Thoralf Skolem (1887–1963) simplified
Löwenheim's proof by introducing Skolem normal forms, and in 1928
Skolem replaced his use of normal forms with a simpler idea, namely to
use what are now called Skolem functions. He used these functions to
convert first-order sentences into universal sentences, that is to
say, into sentences in prenex form with all quantifiers being
universal (\(\forall\)).

Skolem was strongly influenced by Schröder's Algebra der
Logik, starting with his PhD Thesis. Later he took a particular
interest in the quest for an Elimination Theorem in the calculus of
classes. In his 1919 paper he established some results for lattices,
in particular, he showed that one could decide the validity of
universal Horn sentences (i.e., universal sentences with a matrix that
is a disjunction of negated and unnegated atoms, with at most one
positive atom) by a procedure that we now recognize to be a polynomial
time algorithm. This algorithm was based on finding a least fixed
point of a finite partial lattice under production rules derived from
universal Horn sentences. Although this result, which is equivalent to
the uniform word problem for lattices, was in the same paper as
Skolem's famous contribution to Löwenheim's Theorem, it was forgotten
until a chance rediscovery in the early 1990s. (Whitman (1941) gave a
different solution to the more limited equational decision problem for
lattices; it became widely known as the solution to the word problem
in lattices.)

Skolem (1920) gave an elegant solution to the Elimination Problem
posed by Schröder for the calculus of classes by showing that if one
added predicates to express “has at least \(n\) elements”,
for each \(n = 1, 2, \ldots\), then there was a simple (but often
lengthy) procedure to convert a first-order formula about classes into
a quantifier-free formula. In particular this showed that the
first-order theory of the calculus of classes was decidable. This
quantifier-elimination result was used by Mostowski (1952) to analyze
first-order properties of direct powers and direct sums of single
structures, and then by Feferman and Vaught (1959) to do the same for
general direct sums and direct products of structures.

The elimination of quantifiers became a main method in mathematical
logic to prove decidability, and proving decidability was stated as
the main problem of mathematical logic in Hilbert and Ackermann
(1928)—this goal was dropped in subsequent editions because of
the famous undecidability result of Church and Turing.

Model theory can be regarded as the product of Hilbert's
methodology of metamathematics and the algebra of logic tradition,
represented specifically by the results due to Löwenheim and
Skolem. But it was Tarski who gave the discipline its classical
foundation. Model theory is the study of the relations between a
formal language and its interpretation in “realizations”
(that is, a domain for the variables of the language together with an
interpretation for its primitive signs). If the interpretation happens
to make a sentence of the language state something true, then the
interpretation is a model of the sentence (see the
entry on model theory). Models consist
basically of algebraic structures, and model theory became an
autonomous mathematical discipline with its roots not only in the
algebra of logic but in abstract algebra (see Sinaceur
1999).

Apart from model theory, Tarski revived the algebra of relations in
his 1941 paper “On the Calculus of Relations”. First he
outlined a formal logic based on allowing quantification over both
elements and relations, and then he turned to a more detailed study of
the quantifier-free formulas of this system that involved only
relation variables. After presenting a list of axioms that obviously
held in the algebra of relations as presented in Schröder's third
volume he proved that these axioms allowed one to reduce
quantifier-free relation formulas to equations. Thus his calculus of
relations became the study of a certain equational theory which he
noted had the same relation to the study of all binary relations on
sets as the equational theory of Boolean algebra had to the study of
all subsets of sets. This led to questions paralleling those already
posed and resolved for Boolean algebras, for example, was every model
of his axioms for relation algebras isomorphic to an algebra of
relations on a set? One question had been answered by Arwin Korselt
(1864–1947), namely there were first-order sentences in the
theory of binary relations that were not equivalent to an equation in
the calculus of relations—thus the calculus of relations
definitely had a weaker expressive power than the first-order theory
of relations. Actually the expressive power of relation algebra is
exactly equivalent to first-order logic with just three
variables. However, if in relation algebras (the calculus of
relations) one wants to formalize a set theory which has something
such as the pair axiom, then one can reduce many variables to three
variables, and so it is possible to express any first-order statement
of such a theory by an equation. Monk proved that, unlike the calculus of classes,
there is no finite equational basis for the calculus of binary
relations (see Monk 1964). Tarski and Givant (1987) showed that the equational logic
of relation algebras is so expressive that one can carry out
first-order set theory in it.

Furthermore, cylindric algebras, essentially Boolean algebras
equipped with unary cylindric operations \(C_x\) which are intended to
capture the existential quantifiers (\(\exists x\)), were introduced in
the years 1948–1952 by Tarski, working with his students Louise
Chin and Frederick Thompson (see Henkin & Tarski 1961), to create
an algebra of logic that captured the expressive power of the
first-order theory of binary relations. Polyadic algebra is another
approach to an algebra of logic for first-order logic—it was
created by Halmos (1956c). The focus of work in these systems was again to see to
what extent one could parallel the famous results of Stone for Boolean
algebra from the 1930s.

Huntington, E.V., 1904, “Sets of independent postulates for
the algebra of logic”, Transactions of the American
Mathematical Society, 5: 288–309.

–––, 1933, “New sets of independent
postulates for the algebra of logic, with special reference to
Whitehead and Russell's Principia Mathematica”, Transactions
of the American Mathematical Society, 35(1): 274–304.

Jevons, W.S., 1869, The Substitution of Similars, the True
Principle of Reasoning, Derived from a Modification of Aristotle's
Dictum, London: Macmillan and Co.

Peirce, C.S., 1870, “Description of a notation for the logic
of relatives, resulting from an amplification of the conceptions of
Boole's calculus of logic”, Memoirs of the American
Academy, 9: 317–378; reprinted in Collected Papers
1933: Volume III, 27–98.

Wiener, N., 1913, A Comparison between the treatment of the
algebra of relatives by Schroeder and that by Whitehead and
Russell, Ph.D. thesis, Harvard University (Norbert Wiener
Papers. MC 22. Institute Archives and Special Collections, MIT
Libraries, Cambridge, Massachusetts).